block matrix#Block diagonal matrix

{{Short description|Matrix defined using smaller matrices called blocks}}

In mathematics, a block matrix or a partitioned matrix is a matrix that is interpreted as having been broken into sections called blocks or submatrices.{{cite book |last=Eves |first=Howard |author-link=Howard Eves |title=Elementary Matrix Theory |year=1980 |publisher=Dover |location=New York |isbn=0-486-63946-0 |page=[https://archive.org/details/elementarymatrix0000eves_r2m2/page/37 37] |url=https://archive.org/details/elementarymatrix0000eves_r2m2 |url-access=registration |edition=reprint |access-date=24 April 2013 |quote=We shall find that it is sometimes convenient to subdivide a matrix into rectangular blocks of elements. This leads us to consider so-called partitioned, or block, matrices.}}{{Cite web |last=Dobrushkin |first=Vladimir |date= |title=Partition Matrices |url=https://www.cfm.brown.edu/people/dobrush/cs52/Mathematica/Part2/partition.html |access-date=2024-03-24 |website=Linear Algebra with Mathematica}}

Intuitively, a matrix interpreted as a block matrix can be visualized as the original matrix with a collection of horizontal and vertical lines, which break it up, or partition it, into a collection of smaller matrices.{{cite book |last=Anton |first=Howard |title=Elementary Linear Algebra |year=1994 |publisher=John Wiley |location=New York |isbn=0-471-58742-7 |page=30 |edition=7th |quote=A matrix can be subdivided or partitioned into smaller matrices by inserting horizontal and vertical rules between selected rows and columns.}} For example, the 3x4 matrix presented below is divided by horizontal and vertical lines into four blocks: the top-left 2x3 block, the top-right 2x1 block, the bottom-left 1x3 block, and the bottom-right 1x1 block.

: $\left[
\begin{array}{ccc|c}
a_{11} & a_{12} & a_{13} & b_{1} \\
a_{21} & a_{22} & a_{23} & b_{2} \\
\hline
c_{1} & c_{2} & c_{3} & d
\end{array}
\right]$

Any matrix may be interpreted as a block matrix in one or more ways, with each interpretation defined by how its rows and columns are partitioned.

This notion can be made more precise for an $n$ by $m$ matrix $M$ by partitioning $n$ into a collection $\text{rowgroups}$ , and then partitioning $m$ into a collection $\text{colgroups}$ . The original matrix is then considered as the "total" of these groups, in the sense that the $(i, j)$ entry of the original matrix corresponds in a 1-to-1 way with some $(s, t)$ offset entry of some $(x,y)$ , where $x \in \text{rowgroups}$ and $y \in \text{colgroups}$ .{{Cite journal |last1=Indhumathi |first1=D. |last2=Sarala |first2=S. |date=2014-05-16 |title=Fragment Analysis and Test Case Generation using F-Measure for Adaptive Random Testing and Partitioned Block based Adaptive Random Testing |url=http://research.ijcaonline.org/volume93/number6/pxc3895662.pdf |journal=International Journal of Computer Applications |volume=93 |issue=6 |pages=13 |doi=10.5120/16218-5662|bibcode=2014IJCA...93f..11I }}

Block matrix algebra arises in general from biproducts in categories of matrices.{{cite journal | last1 = Macedo | first1 = H.D. | last2 = Oliveira | first2 = J.N. | year = 2013 | title = Typing linear algebra: A biproduct-oriented approach | doi = 10.1016/j.scico.2012.07.012 | journal = Science of Computer Programming | volume = 78 | issue = 11| pages = 2160–2191 | arxiv = 1312.4818 }}

File:BlockMatrix168square.png

Example

The matrix

: $\mathbf{P} = \begin{bmatrix}
1 & 2 & 2 & 7 \\
1 & 5 & 6 & 2 \\
3 & 3 & 4 & 5 \\
3 & 3 & 6 & 7
\end{bmatrix}$

can be visualized as divided into four blocks, as

: $\mathbf{P} = \left[
\begin{array}{cc|cc}
1 & 2 & 2 & 7 \\
1 & 5 & 6 & 2 \\
\hline
3 & 3 & 4 & 5 \\
3 & 3 & 6 & 7
\end{array}
\right]$ .

The horizontal and vertical lines have no special mathematical meaning,{{Cite book |last=Johnston |first=Nathaniel |title=Advanced linear and matrix algebra |date=2021 |publisher=Springer Nature |isbn=978-3-030-52814-0 |location=Cham, Switzerland |pages=298}} but are a common way to visualize a partition. By this partition, $P$ is partitioned into four 2×2 blocks, as

: $\mathbf{P}_{11} = \begin{bmatrix}
1 & 2 \\
1 & 5
\end{bmatrix},\quad
\mathbf{P}_{12} = \begin{bmatrix}
2 & 7\\
6 & 2
\end{bmatrix},\quad
\mathbf{P}_{21} = \begin{bmatrix}
3 & 3 \\
3 & 3
\end{bmatrix},\quad
\mathbf{P}_{22} = \begin{bmatrix}
4 & 5 \\
6 & 7
\end{bmatrix}.$

The partitioned matrix can then be written as

: $\mathbf{P} = \begin{bmatrix}
\mathbf{P}_{11} & \mathbf{P}_{12} \\
\mathbf{P}_{21} & \mathbf{P}_{22}
\end{bmatrix}.$ {{Cite book |last=Jeffrey |first=Alan |url=https://www.worldcat.org/title/639165077 |title=Matrix operations for engineers and scientists: an essential guide in linear algebra |date=2010 |publisher=Springer |isbn=978-90-481-9273-1 |location=Dordrecht [Netherlands] ; New York |pages=54 |oclc=639165077}}

Formal definition

Let $A \in \mathbb{C}^{m \times n}$ . A partitioning of $A$ is a representation of $A$ in the form

: $A = \begin{bmatrix}
A_{11} & A_{12} & \cdots & A_{1q} \\
A_{21} & A_{22} & \cdots & A_{2q} \\
\vdots & \vdots & \ddots & \vdots \\
A_{p1} & A_{p2} & \cdots & A_{pq}
\end{bmatrix}$ ,

where $A_{ij} \in \mathbb{C}^{m_i \times n_j}$ are contiguous submatrices, $\sum_{i=1}^{p} m_i = m$ , and $\sum_{j=1}^{q} n_j = n$ .{{Cite book |last=Stewart |first=Gilbert W. |title=Matrix algorithms. 1: Basic decompositions |date=1998 |publisher=Soc. for Industrial and Applied Mathematics |isbn=978-0-89871-414-2 |location=Philadelphia, PA |pages=18–20}} The elements $A_{ij}$ of the partition are called blocks.

By this definition, the blocks in any one column must all have the same number of columns. Similarly, the blocks in any one row must have the same number of rows.

= Partitioning methods =

A matrix can be partitioned in many ways. For example, a matrix $A$ is said to be partitioned by columns if it is written as

: $A = (a_1 \ a_2 \ \cdots \ a_n)$ ,

where $a_j$ is the $j$ th column of $A$ . A matrix can also be partitioned by rows:

: $A = \begin{bmatrix}
a_1^T \\
a_2^T \\
\vdots \\
a_m^T
\end{bmatrix}$ ,

where $a_i^T$ is the $i$ th row of $A$ .

= Common partitions =

Often, we encounter the 2x2 partition

: $A = \begin{bmatrix}
A_{11} & A_{12} \\
A_{21} & A_{22}
\end{bmatrix}$ ,

particularly in the form where $A_{11}$ is a scalar:

: $A = \begin{bmatrix}
a_{11} & a_{12}^T \\
a_{21} & A_{22}
\end{bmatrix}$ .

Block matrix operations

=Transpose=

Let

: $A = \begin{bmatrix}
A_{11} & A_{12} & \cdots & A_{1q} \\
A_{21} & A_{22} & \cdots & A_{2q} \\
\vdots & \vdots & \ddots & \vdots \\
A_{p1} & A_{p2} & \cdots & A_{pq}
\end{bmatrix}$

where $A_{ij} \in \mathbb{C}^{k_i \times \ell_j}$ . (This matrix $A$ will be reused in {{section link||Addition}} and {{section link||Multiplication}}.) Then its transpose is

: $A^T = \begin{bmatrix}
A_{11}^T & A_{21}^T & \cdots & A_{p1}^T \\
A_{12}^T & A_{22}^T & \cdots & A_{p2}^T \\
\vdots & \vdots & \ddots & \vdots \\
A_{1q}^T & A_{2q}^T & \cdots & A_{pq}^T
\end{bmatrix}$ ,

and the same equation holds with the transpose replaced by the conjugate transpose.

==Block transpose==

A special form of matrix transpose can also be defined for block matrices, where individual blocks are reordered but not transposed. Let $A=(B_{ij})$ be a $k \times l$ block matrix with $m \times n$ blocks $B_{ij}$ , the block transpose of $A$ is the $l \times k$ block matrix $A^\mathcal{B}$ with $m \times n$ blocks $\left(A^\mathcal{B}\right)_{ij} = B_{ji}$ .{{cite thesis |last=Mackey |first=D. Steven |date=2006 |title=Structured linearizations for matrix polynomials |publisher=University of Manchester |issn=1749-9097 |oclc=930686781 |url=http://eprints.maths.manchester.ac.uk/314/1/mackey06.pdf}} As with the conventional trace operator, the block transpose is a linear mapping such that $(A + C)^\mathcal{B} = A^\mathcal{B} + C^\mathcal{B}$ . However, in general the property $(A C)^\mathcal{B} = C^\mathcal{B} A^\mathcal{B}$ does not hold unless the blocks of $A$ and $C$ commute.

=Addition=

Let

: $B = \begin{bmatrix}
B_{11} & B_{12} & \cdots & B_{1s} \\
B_{21} & B_{22} & \cdots & B_{2s} \\
\vdots & \vdots & \ddots & \vdots \\
B_{r1} & B_{r2} & \cdots & B_{rs}
\end{bmatrix}$ ,

where $B_{ij} \in \mathbb{C}^{m_i \times n_j}$ , and let $A$ be the matrix defined in {{section link||Transpose}}. (This matrix $B$ will be reused in {{section link||Multiplication}}.) Then if $p = r$ , $q = s$ , $k_i = m_i$ , and $\ell_j = n_j$ , then

: $A + B = \begin{bmatrix}
A_{11} + B_{11} & A_{12} + B_{12} & \cdots & A_{1q} + B_{1q} \\
A_{21} + B_{21} & A_{22} + B_{22} & \cdots & A_{2q} + B_{2q} \\
\vdots & \vdots & \ddots & \vdots \\
A_{p1} + B_{p1} & A_{p2} + B_{p2} & \cdots & A_{pq} + B_{pq}
\end{bmatrix}$ .

=Multiplication=

It is possible to use a block partitioned matrix product that involves only algebra on submatrices of the factors. The partitioning of the factors is not arbitrary, however, and requires "conformable partitions"{{cite book |last=Eves |first=Howard |author-link=Howard Eves |title=Elementary Matrix Theory |year=1980 |publisher=Dover |location=New York |isbn=0-486-63946-0 |page=[https://archive.org/details/elementarymatrix0000eves_r2m2/page/37 37] |url=https://archive.org/details/elementarymatrix0000eves_r2m2 |url-access=registration |edition=reprint |access-date=24 April 2013 |quote=A partitioning as in Theorem 1.9.4 is called a conformable partition of A and B.}} between two matrices $A$ and $B$ such that all submatrix products that will be used are defined.{{cite book |last=Anton |first=Howard |title=Elementary Linear Algebra |year=1994 |publisher=John Wiley |location=New York |isbn=0-471-58742-7 |page=36 |edition=7th |quote=...provided the sizes of the submatrices of A and B are such that the indicated operations can be performed.}}

{{Cquote

| quote = Two matrices $A$ and $B$ are said to be partitioned conformally for the product $AB$ , when $A$ and $B$ are partitioned into submatrices and if the multiplication $AB$ is carried out treating the submatrices as if they are scalars, but keeping the order, and when all products and sums of submatrices involved are defined.

| author = Arak M. Mathai and Hans J. Haubold

| source = Linear Algebra: A Course for Physicists and Engineers{{Cite book |last1=Mathai |first1=Arakaparampil M. |title=Linear Algebra: a course for physicists and engineers |last2=Haubold |first2=Hans J. |date=2017 |publisher=De Gruyter |isbn=978-3-11-056259-0 |series=De Gruyter textbook |location=Berlin Boston |pages=162}}

}}

Let $A$ be the matrix defined in {{section link||Transpose}}, and let $B$ be the matrix defined in {{section link||Addition}}. Then the matrix product

: $C = AB$

can be performed blockwise, yielding $C$ as an $(p \times s)$ matrix. The matrices in the resulting matrix $C$ are calculated by multiplying:

: $C_{ij} = \sum_{k=1}^{q} A_{ik}B_{kj}.$ {{Cite book |last=Johnston |first=Nathaniel |title=Introduction to linear and matrix algebra |date=2021 |publisher=Springer Nature |isbn=978-3-030-52811-9 |location=Cham, Switzerland |pages=30,425}}

Or, using the Einstein notation that implicitly sums over repeated indices:

: $C_{ij} = A_{ik}B_{kj}.$

Depicting $C$ as a matrix, we have

: $C = AB = \begin{bmatrix}
\sum_{i=1}^{q} A_{1i}B_{i1} & \sum_{i=1}^{q} A_{1i}B_{i2} & \cdots & \sum_{i=1}^{q} A_{1i}B_{is} \\
\sum_{i=1}^{q} A_{2i}B_{i1} & \sum_{i=1}^{q} A_{2i}B_{i2} & \cdots & \sum_{i=1}^{q} A_{2i}B_{is} \\
\vdots & \vdots & \ddots & \vdots \\
\sum_{i=1}^{q} A_{pi}B_{i1} & \sum_{i=1}^{q} A_{pi}B_{i2} & \cdots & \sum_{i=1}^{q} A_{pi}B_{is}
\end{bmatrix}$ .

=Inversion{{anchor|Inversion}}=

{{for|more details and derivation using block LDU decomposition|Schur complement}}

If a matrix is partitioned into four blocks, it can be inverted blockwise as follows:

: ${P} = \begin{bmatrix}
{A} & {B} \\
{C} & {D}
\end{bmatrix}^{-1} = \begin{bmatrix}
{A}^{-1} + {A}^{-1}{B}\left({D} - {CA}^{-1}{B}\right)^{-1}{CA}^{-1} &
-{A}^{-1}{B}\left({D} - {CA}^{-1}{B}\right)^{-1} \\
-\left({D}-{CA}^{-1}{B}\right)^{-1}{CA}^{-1} &
\left({D} - {CA}^{-1}{B}\right)^{-1}
\end{bmatrix},$

where A and D are square blocks of arbitrary size, and B and C are conformable with them for partitioning. Furthermore, A and the Schur complement of A in P: {{nowrap|P/A {{=}} D − CA{{sup|−1}}B}} must be invertible.

{{cite book

| last = Bernstein

| first = Dennis

| title = Matrix Mathematics

| publisher = Princeton University Press

| year = 2005

| pages = 44

| isbn = 0-691-11802-7

}}

Equivalently, by permuting the blocks:

: ${P} = \begin{bmatrix}
{A} & {B} \\
{C} & {D}
\end{bmatrix}^{-1} = \begin{bmatrix}
\left({A} - {BD}^{-1}{C}\right)^{-1} &
-\left({A}-{BD}^{-1}{C}\right)^{-1}{BD}^{-1} \\
-{D}^{-1}{C}\left({A} - {BD}^{-1}{C}\right)^{-1} &
\quad {D}^{-1} + {D}^{-1}{C}\left({A} - {BD}^{-1}{C}\right)^{-1}{BD}^{-1}
\end{bmatrix}.$

Here, D and the Schur complement of D in P: {{nowrap|P/D {{=}} A − BD{{sup|−1}}C}} must be invertible.

If A and D are both invertible, then:

: $\begin{bmatrix}
{A} & {B} \\
{C} & {D}
\end{bmatrix}^{-1} = \begin{bmatrix}
\left({A} - {B} {D}^{-1} {C}\right)^{-1} & {0} \\
{0} & \left({D} - {C} {A}^{-1} {B}\right)^{-1}
\end{bmatrix} \begin{bmatrix}
{I} & -{B} {D}^{-1} \\
-{C} {A}^{-1} & {I}
\end{bmatrix}.$

By the Weinstein–Aronszajn identity, one of the two matrices in the block-diagonal matrix is invertible exactly when the other is.

=Determinant{{anchor|Determinant}}=

The formula for the determinant of a $2 \times 2$ -matrix above continues to hold, under appropriate further assumptions, for a matrix composed of four submatrices $A, B, C, D$ with $A$ and $D$ square. The easiest such formula, which can be proven using either the Leibniz formula or a factorization involving the Schur complement, is

: $\det\begin{bmatrix}A& 0\\ C& D\end{bmatrix} = \det(A) \det(D) = \det\begin{bmatrix}A& B\\ 0& D\end{bmatrix}.$

Using this formula, we can derive that characteristic polynomials of $\begin{bmatrix}A& 0\\ C& D\end{bmatrix}$ and $\begin{bmatrix}A& B\\ 0& D\end{bmatrix}$ are same and equal to the product of characteristic polynomials of $A$ and $D$ . Furthermore, If $\begin{bmatrix}A& 0\\ C& D\end{bmatrix}$ or $\begin{bmatrix}A& B\\ 0& D\end{bmatrix}$ is diagonalizable, then $A$ and $D$ are diagonalizable too. The converse is false; simply check $\begin{bmatrix}1& 1\\ 0& 1\end{bmatrix}$ .

If $A$ is invertible, one has

: $\det\begin{bmatrix}A& B\\ C& D\end{bmatrix} = \det(A) \det\left(D - C A^{-1} B\right),$

and if $D$ is invertible, one has

: $\det\begin{bmatrix}A& B\\ C& D\end{bmatrix} = \det(D) \det\left(A - B D^{-1} C\right) .$ Taboga, Marco (2021). "Determinant of a block matrix", Lectures on matrix algebra.

If the blocks are square matrices of the same size further formulas hold. For example, if $C$ and $D$ commute (i.e., $CD=DC$ ), then

: $\det\begin{bmatrix}A& B\\ C& D\end{bmatrix} = \det(AD - BC).$ {{Cite journal|first= J. R.|last= Silvester|title= Determinants of Block Matrices|journal= Math. Gaz.|volume= 84|issue= 501|year= 2000|pages= 460–467|jstor= 3620776|url= http://www.ee.iisc.ernet.in/new/people/faculty/prasantg/downloads/blocks.pdf|doi= 10.2307/3620776|access-date= 2021-06-25|archive-date= 2015-03-18|archive-url= https://web.archive.org/web/20150318222335/http://www.ee.iisc.ernet.in/new/people/faculty/prasantg/downloads/blocks.pdf|url-status= dead}}

This is also true when $AB=BA$ , $AC=CA$ , or {{tmath|1=BD=DB}}.

This formula has been generalized to matrices composed of more than $2 \times 2$ blocks, again under appropriate commutativity conditions among the individual blocks.{{cite journal|last1=Sothanaphan|first1=Nat|title=Determinants of block matrices with noncommuting blocks|journal=Linear Algebra and Its Applications|date=January 2017|volume=512|pages=202–218|doi=10.1016/j.laa.2016.10.004|arxiv=1805.06027|s2cid=119272194}}

For $A = D$ and $B=C$ , the following formula holds (even if $A$ and $B$ do not commute)

: $\det\begin{bmatrix}A& B\\ B& A\end{bmatrix} = \det(A - B) \det(A + B).$

Special types of block matrices

=Direct sums and block diagonal matrices=

==Direct sum==

For any arbitrary matrices A (of size m × n) and B (of size p × q), we have the direct sum of A and B, denoted by A $\oplus$ B and defined as

: ${A} \oplus {B} =
\begin{bmatrix}
a_{11} & \cdots & a_{1n} & 0 & \cdots & 0 \\
\vdots & \ddots & \vdots & \vdots & \ddots & \vdots \\
a_{m1} & \cdots & a_{mn} & 0 & \cdots & 0 \\
0 & \cdots & 0 & b_{11} & \cdots & b_{1q} \\
\vdots & \ddots & \vdots & \vdots & \ddots & \vdots \\
0 & \cdots & 0 & b_{p1} & \cdots & b_{pq}
\end{bmatrix}.$

For instance,

: $\begin{bmatrix}
1 & 3 & 2 \\
2 & 3 & 1
\end{bmatrix} \oplus
\begin{bmatrix}
1 & 6 \\
0 & 1
\end{bmatrix} =
\begin{bmatrix}
1 & 3 & 2 & 0 & 0 \\
2 & 3 & 1 & 0 & 0 \\
0 & 0 & 0 & 1 & 6 \\
0 & 0 & 0 & 0 & 1
\end{bmatrix}.$

This operation generalizes naturally to arbitrary dimensioned arrays (provided that A and B have the same number of dimensions).

Note that any element in the direct sum of two vector spaces of matrices could be represented as a direct sum of two matrices.

==Block diagonal matrices {{anchor|Block diagonal matrix}} ==

A block diagonal matrix is a block matrix that is a square matrix such that the main-diagonal blocks are square matrices and all off-diagonal blocks are zero matrices.{{Cite book |last1=Abadir |first1=Karim M. |title=Matrix Algebra |last2=Magnus |first2=Jan R. |publisher=Cambridge University Press |year=2005 |isbn=9781139443647 |pages=97,100,106,111,114,118 |language=en}} That is, a block diagonal matrix A has the form

: ${A} = \begin{bmatrix}
{A}_1 & {0} & \cdots & {0} \\
{0} & {A}_2 & \cdots & {0} \\
\vdots & \vdots & \ddots & \vdots \\
{0} & {0} & \cdots & {A}_n
\end{bmatrix}$

where A_k is a square matrix for all k = 1, ..., n. In other words, matrix A is the direct sum of A₁, ..., A_n. It can also be indicated as A₁ ⊕ A₂ ⊕ ... ⊕ A_n or diag(A₁, A₂, ..., A_n){{Cite book |last=Gentle |first=James E. |title=Matrix Algebra: Theory, Computations, and Applications in Statistics |date=2007 |publisher=Springer New York Springer e-books |isbn=978-0-387-70873-7 |series=Springer Texts in Statistics |location=New York, NY |pages=47,487}} (the latter being the same formalism used for a diagonal matrix). Any square matrix can trivially be considered a block diagonal matrix with only one block.

For the determinant and trace, the following properties hold:

: $\begin{align}
\det{A} &= \det{A}_1 \times \cdots \times \det{A}_n,
\end{align}$ {{Cite book |last1=Quarteroni |first1=Alfio |title=Numerical mathematics |last2=Sacco |first2=Riccardo |last3=Saleri |first3=Fausto |date=2000 |publisher=Springer |isbn=978-0-387-98959-4 |series=Texts in applied mathematics |location=New York |pages=10,13}}{{Cite journal |last1=George |first1=Raju K. |last2=Ajayakumar |first2=Abhijith |date=2024 |title=A Course in Linear Algebra |url=https://doi.org/10.1007/978-981-99-8680-4 |journal=University Texts in the Mathematical Sciences |language=en |pages=35,407 |doi=10.1007/978-981-99-8680-4 |isbn=978-981-99-8679-8 |issn=2731-9318}} and

: $\begin{align}
\operatorname{tr}{A} &= \operatorname{tr} {A}_1 + \cdots + \operatorname{tr} {A}_n.\end{align}$

A block diagonal matrix is invertible if and only if each of its main-diagonal blocks are invertible, and in this case its inverse is another block diagonal matrix given by

: $\begin{bmatrix}
{A}_{1} & {0} & \cdots & {0} \\
{0} & {A}_{2} & \cdots & {0} \\
\vdots & \vdots & \ddots & \vdots \\
{0} & {0} & \cdots & {A}_{n}
\end{bmatrix}^{-1} = \begin{bmatrix}
{A}_{1}^{-1} & {0} & \cdots & {0} \\
{0} & {A}_{2}^{-1} & \cdots & {0} \\
\vdots & \vdots & \ddots & \vdots \\
{0} & {0} & \cdots & {A}_{n}^{-1}
\end{bmatrix}.$ {{Cite book |last=Prince |first=Simon J. D. |title=Computer vision: models, learning, and inference |date=2012 |publisher=Cambridge university press |isbn=978-1-107-01179-3 |location=New York |pages=531}}

The eigenvalues and eigenvectors of ${A}$ are simply those of the ${A}_k$ s combined.

=Block tridiagonal matrices=

A block tridiagonal matrix is another special block matrix, which is just like the block diagonal matrix a square matrix, having square matrices (blocks) in the lower diagonal, main diagonal and upper diagonal, with all other blocks being zero matrices. It is essentially a tridiagonal matrix but has submatrices in places of scalars. A block tridiagonal matrix $A$ has the form

: ${A} = \begin{bmatrix}
{B}_{1} & {C}_{1} & & & \cdots & & {0} \\
{A}_{2} & {B}_{2} & {C}_{2} & & & & \\
& \ddots & \ddots & \ddots & & & \vdots \\
& & {A}_{k} & {B}_{k} & {C}_{k} & & \\
\vdots & & & \ddots & \ddots & \ddots & \\
& & & & {A}_{n-1} & {B}_{n-1} & {C}_{n-1} \\
{0} & & \cdots & & & {A}_{n} & {B}_{n}
\end{bmatrix}$

where ${A}_{k}$ , ${B}_{k}$ and ${C}_{k}$ are square sub-matrices of the lower, main and upper diagonal respectively.{{Cite book |last=Dietl |first=Guido K. E. |url=https://www.worldcat.org/title/ocm85898525 |title=Linear estimation and detection in Krylov subspaces |date=2007 |publisher=Springer |isbn=978-3-540-68478-7 |series=Foundations in signal processing, communications and networking |location=Berlin ; New York |pages=85,87 |language=en |oclc=ocm85898525}}{{Cite book |last1=Horn |first1=Roger A. |title=Matrix analysis |last2=Johnson |first2=Charles R. |date=2017 |publisher=Cambridge University Press |isbn=978-0-521-83940-2 |edition=Second edition, corrected reprint |location=New York, NY |pages=36 |language=en}}

Block tridiagonal matrices are often encountered in numerical solutions of engineering problems (e.g., computational fluid dynamics). Optimized numerical methods for LU factorization are available{{Cite book |last=Datta |first=Biswa Nath |title=Numerical linear algebra and applications |date=2010 |publisher=SIAM |isbn=978-0-89871-685-6 |edition=2 |location=Philadelphia, Pa |pages=168}} and hence efficient solution algorithms for equation systems with a block tridiagonal matrix as coefficient matrix. The Thomas algorithm, used for efficient solution of equation systems involving a tridiagonal matrix can also be applied using matrix operations to block tridiagonal matrices (see also Block LU decomposition).

=Block triangular matrices=

==Upper block triangular==

A matrix $A$ is upper block triangular (or block upper triangular{{Cite book |last=Stewart |first=Gilbert W. |title=Matrix algorithms. 2: Eigensystems |date=2001 |publisher=Soc. for Industrial and Applied Mathematics |isbn=978-0-89871-503-3 |location=Philadelphia, Pa |pages=5}}) if

: $A = \begin{bmatrix}
A_{11} & A_{12} & \cdots & A_{1k} \\
0 & A_{22} & \cdots & A_{2k} \\
\vdots & \vdots & \ddots & \vdots \\
0 & 0 & \cdots & A_{kk}
\end{bmatrix}$ ,

where $A_{ij} \in \mathbb{F}^{n_i \times n_j}$ for all $i, j = 1, \ldots, k$ .{{Cite book |last=Bernstein |first=Dennis S. |title=Matrix mathematics: theory, facts, and formulas |publisher=Princeton University Press |year=2009 |isbn=978-0-691-14039-1 |edition=2 |location=Princeton, NJ |pages=168,298 |language=en}}

==Lower block triangular==

A matrix $A$ is lower block triangular if

: $A = \begin{bmatrix}
A_{11} & 0 & \cdots & 0 \\
A_{21} & A_{22} & \cdots & 0 \\
\vdots & \vdots & \ddots & \vdots \\
A_{k1} & A_{k2} & \cdots & A_{kk}
\end{bmatrix}$ ,

where $A_{ij} \in \mathbb{F}^{n_i \times n_j}$ for all $i, j = 1, \ldots, k$ .

=Block Toeplitz matrices=

A block Toeplitz matrix is another special block matrix, which contains blocks that are repeated down the diagonals of the matrix, as a Toeplitz matrix has elements repeated down the diagonal.

A matrix $A$ is block Toeplitz if $A_{(i,j)} = A_{(k,l)}$ for all $k - i = l - j$ , that is,

: $A = \begin{bmatrix}
A_1 & A_2 & A_3 & \cdots \\
A_4 & A_1 & A_2 & \cdots \\
A_5 & A_4 & A_1 & \cdots \\
\vdots & \vdots & \vdots & \ddots
\end{bmatrix}$ ,

where $A_i \in \mathbb{F}^{n_i \times m_i}$ .

=Block Hankel matrices=

A matrix $A$ is block Hankel if $A_{(i,j)} = A_{(k,l)}$ for all $i + j = k + l$ , that is,

: $A = \begin{bmatrix}
A_1 & A_2 & A_3 & \cdots \\
A_2 & A_3 & A_4 & \cdots \\
A_3 & A_4 & A_5 & \cdots \\
\vdots & \vdots & \vdots & \ddots
\end{bmatrix}$ ,